commit ec6143d8dd2255580da4ad609adab157781ce0ad
parent f34939dba033645cad998cbf3e1dd8d361922c62
Author: Andrew <andrewlaack1@gmail.com>
Date: Tue, 13 Aug 2024 18:07:13 -0500
Took notes on stats+lin-alg+discrete math
Diffstat:
12 files changed, 145 insertions(+), 1 deletion(-)
diff --git a/Biconditional.md b/Biconditional.md
@@ -0,0 +1,19 @@
+:discrete:
+# Biconditional (iff)
+
+1.1.2
+
+## Notes
+
+**Definition:** The biconditional is the [[Connectives.md]] that states the antecedent and consequent have the same truth values.
+
+$p \iff q$ this can be stated as p iff q, if and only if p then q, or some other way.
+
+Basically, this is only true when both propositions share the same truth value.
+
+| p | q | $p \iff q$ |
+|---|---|------------|
+| T | T | T |
+| T | F | F |
+| F | T | F |
+| F | F | T |
diff --git a/Connectives.md b/Connectives.md
@@ -1,5 +1,5 @@
:discrete:
-# Connectives
+# Connectives (Logical Operators)
1.1.1
@@ -13,3 +13,17 @@
- Implication (If, then | $\to$)
- Biconditional (If and only if | $\iff$)
- Exclusive or ($\oplus$)
+
+Simple Proposition:
+
+The sun is red.
+
+Compound Proposition:
+
+The sun is red on tuesday.
+
+p = tuesday
+
+q = sun is red
+
+$p \to q$
diff --git a/Contrapositive.md b/Contrapositive.md
@@ -6,3 +6,7 @@ Throughout TB
## Notes
**Definition:** To prove and if then statement with contrapositive we assume the then statement is false. Following from here we then prove the if part must also be true for the then to be false. So it follows that if the first is true then the second is also true because the second is never true when the first is false.
+
+This is of the form $\neg q \to \neq p$ where we switch the statements and negate both. To just negate both we [[Inverse.md]] it.
+
+This always has the same truth value as the original.
diff --git a/Converse.md b/Converse.md
@@ -0,0 +1,10 @@
+:discrete:
+# Converse
+
+1.1.2
+
+## Notes
+
+**Definition:** The converse of a statement is to switch both sides of an implication statement.
+
+$p \to q$ converse is $q \to p$.
diff --git a/DiscreteMath.md b/DiscreteMath.md
@@ -11,3 +11,7 @@ Unit 1.1:
- [[Proposition.md]]
- [[Negation.md]]
- [[Connectives.md]]
+ - [[Converse.md]] - Switch both
+ - [[Inverse.md]] - Negate both
+ - [[Contrapositive.md]] - Swap then inverse
+ - [[Biconditional.md]] - IFF
diff --git a/Inverse.md b/Inverse.md
@@ -0,0 +1,10 @@
+:discrete:
+# Inverse
+
+1.1.2
+
+## Notes
+
+**Definition:** The inverse of an implication statement is the negation of both terms.
+
+$\neg p \to \neg q$ Where the original was $p \to q$
diff --git a/LawOfLargeNumbers.md b/LawOfLargeNumbers.md
@@ -0,0 +1,10 @@
+:prob:
+# Law of Large Numbers (LLN)
+
+L19
+
+## Notes
+
+**Definition:** The average results from a large set of independent trials converges upon the true value.
+
+See also [[RegressionToTheMean.md]]
diff --git a/LinearAlgebra.md b/LinearAlgebra.md
@@ -71,3 +71,4 @@ Khan Unit 2:
- [[CramersRule.md]]
- [[GaussianElimination.md]]
- [[EigenVector.md]]
+ - [[Transpose.md]]
diff --git a/MarkovInequality.md b/MarkovInequality.md
@@ -0,0 +1,8 @@
+:prob:
+# Markov Inequality
+
+L19
+
+## Notes
+
+**Definition:** The Markov inequality gives the probability that a random variable is greater than or equal to some constant.
diff --git a/RegressionToTheMean.md b/RegressionToTheMean.md
@@ -0,0 +1,8 @@
+:prob:
+# Regression to the Mean
+
+L19
+
+## Notes
+
+**Definition:** Regression to the mean is the idea that if an unlikely event occurs it is likely the next sampling will be closer to the mean of the distribution.
diff --git a/StatisticsAndProbability.md b/StatisticsAndProbability.md
@@ -102,3 +102,9 @@ L16:
L17:
- [[MarkovChains.md]]
- [[PeriodicChain.md]]
+L18:
+ - Skipped markov part 3
+L19:
+ - [[LawOfLargeNumbers.md]]
+ - [[RegressionToTheMean.md]]
+ - [[MarkovInequality.md]]
diff --git a/Transpose.md b/Transpose.md
@@ -14,3 +14,53 @@ ML P627
As you can see, the first value remains and across the top we have the first column.
Additionally, the transpose of a vector is possible and will go from n x 1 to 1 x n.
+
+Example:
+
+ [1 8 0]
+A = [7 6 4]
+ [2 1 6]
+
+ [1 7 2]
+A^T = [8 6 1]
+ [0 4 6]
+
+
+B = [1 2]
+ [3 4]
+
+B^T = [1 3]
+ [2 4]
+
+C = [1 0 -1]
+ [2 7 -5]
+ [4 -3 2]
+ [-1 3 0]
+
+C^T = [ 1 2 4 -1]
+ [ 0 7 -3 3]
+ [-1 -5 2 0]
+
+Also, note that (C^T)^T = C.
+
+#### Determinant
+
+The determinant of a matrix's transpose is the same as the determinant prior to the transpose. |A| = |A^T|
+
+Example:
+
+B = [1 2]
+ [3 4]
+
+|B| = 4 - 6 = -2
+
+B^T = [1 3]
+ [2 4]
+
+|B^T| = 4 - 6 = -2
+
+#### Product of transpose
+
+(AB)^T = B^T A^T
+
+Note: This can be scaled up with an arbitrarily large list of matricies.
